When \(n=0\) , as explained in [1]}, the stack \(\overline{\mathcal {J}}_{g,0}(\phi ^{can})\)  is isomorphic to Caporaso's universal compactified Jacobian [2]}, [3]}, which is studied in detail in [4]}, [5]}, [6]}, [7]}, [6]}, [9]}, [10]}, [11]} and is, in turn, isomorphic to Pandharipande's universal compactified Jacobian [12]} (see [13]}).
Note that in this case \(\phi ^{can}\)  is general precisely when \(d-g+1\)  and \(2g-2\)  are coprime (see [2]} and [1]}).
